If \(\operatorname{depth}G(\mathfrak {m} )\ge 1,\)  then the conclusion follows from [1]}.
Suppose \(\operatorname{depth}G(\mathfrak {m} )=0.\)  By [2]}, \(\widetilde{\mathfrak {m} ^j}=\mathfrak {m} ^j\)  for \(j\ge 3\)  which implies \(\operatorname{depth}G(\mathfrak {m} ^3)>0.\)  Then by Theorem REF , \(r_J(\mathfrak {m} )\le e_1(\mathfrak {m} )-e_0(\mathfrak {m} )+\lambda (R/\mathfrak {m} )+3\) .
